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32a review

# 32a review - 13 Three-Dimensional Coordinate Systems I hope...

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13. Three-Dimensional Coordinate Systems: - I hope you guys know this by heart already... there's a distance formula, and not much else. 13.2 Vectors: - This section is also pretty basic, and lays out vector addition and scalar multiplication. You guys should know this pretty well too... 13.3 The Dot Product: - For the underinformed, the dot product gives a SCALAR value, not a vector value. i.e. <(1,2,3),(3,2,1)> = (1)(3)+(2)(2)+(3)(1) = 10 - The formula for the angle between two vectors, cos(theta) = a.b/(|a||b|), is useful, both in computing a.b and the angle theta. - The formulas for scalar and vector projections are probably less fundamental (in my opinion), but as always, if you have time, it can be useful to know these formulas. 13.4 The Cross Product - The cross product of two vectors, axb, gives a vector orthogonal to both a and b via the right hand rule. This is, in my opinion, trickier than the dot product, and it is important to briefly remind yourself about the basic properties (i.e. bxa = -axb, etc.) - (trick): Two vectors a, b are parallel iff axb = 0. Sometimes it is easier to just compute the cross product than to try to find lambda with a = lambda*b. - The scalar triple product ((axb).c) = det([a b c]) and the formula sin(theta) = |axb|/(|a||b|) are good to know, but probably less fundamental than other cross product facts, and the sin(theta) formula is not used as much as the cos(theta) formula. 13.5 Equations of Lines and Planes - A point and a normal vector uniquely determine a plane; - Three (non-colinear) points uniquely determine a plane; - A line and a point (not lying on that line) uniquely determine a plane; - An important form for the equation of a plane is <v,((x,y,z)-p)> = 0, where p is any point on the plane, and v is the normal vector. - There are two forms for an equation of a line: the parametric equations (more useful, I think), and the symmetric equations. 13.6 Cylinders and Quadric Surfaces - The table on p.872 is pretty useful, and worth memorizing. Break down the quadrics into ones with constants in the equation (ellipsoid, hyperboloids of (one and two) sheets, cylinder), and without (cone, (elliptic and hyperbolic) paraboloids). - As an example of a possible mnemonic to remember the surfaces further, all of the non-paraboloid surfaces have all quadratic terms (x^2,y^2,z^2). - The traces (most of the time) will give you the name of the surface.

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- For the equations with constants in them, the number of minus signs (in standard form) completely classifies the surface. 14.1 Vector Functions and Space Curves - The limit and derivative of a vector-valued function can be taken componentwise, i.e.
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32a review - 13 Three-Dimensional Coordinate Systems I hope...

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